Basic commands using the "Insert" menu: To insert a two-dimensional (2D) graph, use: To insert a three-dimensional (3D) graph, use: Insert > Plot > 3D

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1 Oct 7::3 - GraphsBasics5_ForPrinting.sm Eamples of two- and three-dimensional graphics in Smath Studio B Gilberto E. Urro, October Basic commands using the "Insert" menu: To insert a two-dimensional (D) graph, use: To insert a three-dimensional (3D) graph, use: Insert > Plot > D Insert > Plot > 3D EXAMPLE A - Plotting a single function of : - Clic on point in our worsheet where ou want to set the upper left corner of graph - Clic on the "D" icon in the "Functions" palette or use the "Insert > Graph > D" menu option 3 - Tpe the function name in the placeholder below the graph. In this eample we plot the function: f() = sin() - - Thus, tpe: "sin()", then clic somewhere in the worsheet outside of the graph. - - sin EXAMPLE B - Plotting a single function of (,): - Clic on point in our worsheet where ou want to set the upper left corner of graph - Clic on the "3D" icon in the "Functions" palette or use the "Insert > Graph > 3D" menu option 3 - Tpe the function name in the placeholder below the graph. In this eample we plot the function: f() = *sin() Thus, tpe: "*sin()", then clic somewhere in the worsheet outside of the graph. sin Using icons in the "Functions" palette: Clic on the "D" or "3D" icon in the palette to insert a D or 3D graph: -> -> -> -> -> -> -> -> Also, use the "Multiple Values" icon -> -> to plot more than one function. EXAMPLE A - Plotting two functions of in D: / 3

2 - Clic on point in our worsheet where ou want to set the upper left corner of graph - Clic on the "D" icon in the "Functions" palette or use the "Insert > Graph > D" menu option 3 - Clic on the placeholder below the graph - Clic on the "Multiple values" icon, then tpe the functions to be plotted in each placeholder in the open brace Oct 7::3 - GraphsBasics5_ForPrinting.sm In this eample we plot the functions: "sin()" and "cos()", then clic somewhere in the worsheet outside of the graph sin cos EXAMPLE B - Plotting two functions of (,) in 3D: - Clic on point in our worsheet where ou want to set the upper left corner of graph - Clic on the "3D" icon in the "Functions" palette or use the "Insert > Graph > 3D" menu option 3 - Clic on the placeholder below the graph - Clic on the "Multiple values" icon, then tpe the functions to be plotted in each placeholder in the open brace In this eample we plot the functions: "*sin()" and "+", then clic somewhere in the worsheet outside of the graph. sin Adding more entries to the "Multiple values" icon: The "Multiple values" icon produces two entries b default. However, ou can insert more entries b following this procedure: / 3

3 Oct 7::3 - GraphsBasics5_ForPrinting.sm () Clic somewhere in the worsheet, then clic on the "Multiple value" icon in the "Functions" palette, to produce the "Multiple values" brace as shown. This figure shows the default case of "Multiple values," i.e., a brace with two entries. () Clic on the left-hand side of the frame of () to produce the holder in the lower right corner as shown. (3) Place the mouse over the holder until the double arrow icon shows up... () then, immediatel drag the holder down to insert a third entr in the "Multiple values" brace. (5) If needed, repeat steps () through () to add more entries to the "Multiple values" brace. EXAMPLE 3A - Plotting three functions of in D: - Clic on point in our worsheet where ou want to set the upper left corner of graph - Clic on the "D" icon in the "Functions" palette or use the "Insert > Graph > D" menu option 3 - Clic on the placeholder below the graph - Clic on the "Multiple values" icon in the "Function" palette 5 - Add one more entr to the "Multiple values" brace. In this eample we plot the functions: "sin()", "cos()", and "sin(*)"then clic somewhere in the worsheet outside of the graph. As ou add functions to plot, SMath Studio uses the following colors for the plots: sin cos sin - blue - red 3 - blac - magenta, etc. EXAMPLE 3B - Plotting three functions of (,) in 3D: 3 / 3

4 Oct 7::3 - GraphsBasics5_ForPrinting.sm - Clic on point in our worsheet where ou want to set the upper left corner of graph - Clic on the "3D" icon in the "Functions" palette or use the "Insert > Graph > 3D" menu option 3 - Clic on the placeholder below the graph - Clic on the "Multiple values" icon in the "Function" palette 5 - Add one more entr to the "Multiple values" brace. In this eample we plot the functions: "*sin()", "+", and "-"then clic somewhere in the worsheet outside of the graph. sin Icons in the "Plot" palette: -> -> -> -> -> -> -> () Rotate: Rotate a 3D graph onl () Scale:(see instructions below) (3) Move: drag graph up or down, left or right () Graph b points: show points instead of lines (5) Graph b lines: show lines (default) () Refresh: restore to original version of graph Detalles of "Scale" in a D graph: * ZOOM IN or OUT: Clic on the "Scale" icon, icon (), then clic on the graph (also for 3D plots): - ZOOM IN: Drag the mouse inwards, towards the origin, to decrease sie of aes divisions - ZOOM OUT: Drag the mouse outwards, awa from the origin, to increase sie of ais divisions * Alternativel, to ZOOM IN or OUT, clic on the "Scale" icon, clic inside the graph and use the mouse wheel (also for 3D plots): - ZOOM IN: roll mouse wheel up - ZOOM OUT: roll mouse wheel down * ZOOM IN or OUT on the -ais onl: Clic on the "Scale" icon, clic on the graph, hold the [SHIFT] e, then: - ZOOM IN X-AXIS: roll mouse wheel up - ZOOM OUT X-AXIS: roll mouse wheel down * ZOOM IN or OUT on the -ais onl: Clic on the "Scale" icon, clic on the graph, hold the [CTRL] e, then: - ZOOM IN Y-AXIS: roll mouse wheel up - ZOOM OUT Y-AXIS: roll mouse wheel down EXAMPLE A - Changing the sie of the graph window in D: Clic on the graph window, then drag one of the three blac handlers in the graph window to adjust its sie / 3

5 Oct 7::3 - GraphsBasics5_ForPrinting.sm () sin (3) sin sin For the graphs above: () Dragging the right-hand side handler to shorten the width, and the bottom handler to enlarge the heigh of the graph. () Dragging the bottom handler to reduce the height of the graph. (3) Dragging the right-bottom corner handler to reduce the sie proportionall EXAMPLE B - Changing the sie of the graph window in 3D: The procedure is ver similar to that of a D graph. sin cos sin cos sin cos EXAMPLE 5A - Moving the aes about the graph window (D): 5 / 3

6 Oct 7::3 - GraphsBasics5_ForPrinting.sm sin -- sin - Clic on the "Move" option in the "Plot" palette - Clic on the graph window and drag the mouse in the direction where ou want to move the aes. EXAMPLE 5B - Moving the aes about the graph window (3D): The procedure is ver similar to that of a D graph. cos sin cos sin Detalles of "Scale" in a D graph (repeated): / 3

7 Oct 7::3 - GraphsBasics5_ForPrinting.sm * ZOOM IN or OUT: Clic on the "Scale" icon, icon (), then clic on the graph (also for 3D plots): - ZOOM IN: Drag the mouse inwards, towards the origin, to decrease sie of aes divisions - ZOOM OUT: Drag the mouse outwards, awa from the origin, to increase sie of ais divisions * Alternativel, to ZOOM IN or OUT, clic on the "Scale" icon, clic inside the graph and use the mouse wheel (also for 3D plots): - ZOOM IN: roll mouse wheel up - ZOOM OUT: roll mouse wheel down * ZOOM IN or OUT on the -ais onl: Clic on the "Scale" icon, clic on the graph, hold the [SHIFT] e, then: - ZOOM IN X-AXIS: roll mouse wheel up - ZOOM OUT X-AXIS: roll mouse wheel down * ZOOM IN or OUT on the -ais onl: Clic on the "Scale" icon, clic on the graph, hold the [CTRL] e, then: - ZOOM IN Y-AXIS: roll mouse wheel up - ZOOM OUT Y-AXIS: roll mouse wheel down EXAMPLE A - Scaling (ooming) a D graph: <--- To oom the -ais onl: - Clic on "Scale" in the "Plot" palette - Clic on the graph window 3 - Hold down the "Shift" e - Roll the mouse wheel up or down sin To oom the -ais onl: ----> - Clic on "Scale" in the "Plot" palette - Clic on the graph window 3 - Hold down the "Control" e - Roll the mouse wheel up or down sin 7 / 3

8 Oct 7::3 - GraphsBasics5_ForPrinting.sm sin <--- You can oom both aes b ooming one ais at a time. In this case, I oomed the ais first, and then the ais. Note: Use the "Refresh" option in the "Plot" menu to recover the original version of an plot. EXAMPLE B - Scaling (ooming) a 3D graph: sin cos sin cos Eamples of other tpes of graphs in -D The following eamples show other was to produce D graphics. Data for a graph = f() can be generated b using a vector of values of, then generating a vector of values of. The two vectors are then put together into a matri, whose name is used in the D graph placeholder instead of f(). EXAMPLE 7 - Plotting a function using vectors: Vectors of and data are created using ranges, eample: n n π π, π.. π length Create vector as follows Tpe: : range - p cntl-g, p cntl-g, - p cntl-g + p cntl-g / Calculate the length of vector = n / 3

9 for.. n sin Oct 7::3 - GraphsBasics5_ForPrinting.sm Fill out vector using a for loop. Clic "for" in the "Programming" palette, then use: range, n sin Use sub-indices, e.g., [... etc. M augment, Form augmented matri M with vectors and, place M in graph as a function name:.5 < --- The graph was oomed in and the aes moved b using the following procedures: M - - To oom -ais onl: clic on "Scale" in the "Plot" palette, hold the "Control" e, and use the mouse wheel - To oom -ais onl: clic on "Scale" in the "Plot" paletted, hold the "Shift" e and use the mouse wheel 3 - To move aes, drag mouse across graph window Using points or lines for a plot:.5.5 Using the sparse data in matri M we reproduce the graph above, but then we selected the "Graph b points" option in the "Plot" palette to produce the graph shown to the left. You can clic the option "Graph b lines" option in the "Plot" palette to return to the default graph format of continuous lines M EXAMPLE - Plotting a function and a matri In this eample we plot the function = cos() and a matri M with vector data of = sin() in the range -π < < π. π π, π.. π for.. n sin n length n M augment, 9 / 3

10 Oct 7::3 - GraphsBasics5_ForPrinting.sm M cos M cos - Original graph - Zooming In in both and EXAMPLE 9 - Parametric plots in D using matrices: Parametric plots are plots of the form = (t), = (t). A parametric plot can be generated b using vectors and matrices as illustrated below. Use a fine grid for the parameter t to produce a continuous curve. π t π, π.. π 5 n length t for.. n sin 3 t cos t Define the vector of the parameter t Determine length of vector t = n Calculate vectors of = (t) and = (t) M augment, Produce matri of (,) and plot it M M Using "Plot b Lines" option in the "Plot" palette - Using "Plot b Lines" option in the "Plot" palette EXAMPLE - Polar plots produced using vectors and matrices: / 3

11 Oct 7::3 - GraphsBasics5_ForPrinting.sm Polar plots are similar to parametric plots. In polar plots the independent variable is the angle θ, and the dependent variable is the radial position r, i.e., r = f(θ). To produce the plot the (,) coordinates are calculated using = r*cos(θ) and = r*sin(θ) as illustrated below. θ n π,.. 5 length θ for r.. n π sin θ Generate vector of θ between and π Determine lenght of vector θ Generate values of r = f(θ) for.. n r cos r sin θ θ Generate coordinates: = r cos(θ) = r sin(θ) P augment, Produce matri of (,) and plot it P P Using "Plot b Lines" option in the "Plot" palette - Using "Plot b Lines" option in the "Plot" palette EXAMPLE - Parametric plot in three-dimensions - space curves Parametric equations of the form = (t), = (t), = (t), produces a space curve. To generate the graph, use a vector of values of t, and then calculate the corresponding vectors of values of,, and. Put together a matri whose columns are the,, data, and use a 3D plot. t n n,... length t Create a vector t with values of the parameter that will produce = (t), = (t), and = (t). Determine the length of vector t / 3

12 for.. n sin t cos t t Oct 7::3 - GraphsBasics5_ForPrinting.sm Generate vectors,, and using a "for" loop M augment,, Build matri M with coordinates (,,) Plot matri M in a 3D plot M EXAMPLE - Using D graphs in solving equations: In this eample we see the solution(s) for the equation: 3 5 A solution can be found b determining the intersection of the functions: 3 f g 5 Using graphics and ooming the intersection we estimate the solution to be close to = f g f g The eact solution can be found using: solve 3 5,.7 / 3

13 Oct 7::3 - GraphsBasics5_ForPrinting.sm EXAMPLE - Graphical solution for a pump-pipeline sstem (Civil Engineering Hdraulics) Solution: ========= Write out all the given data without units, but using the proper set of units for the English Sstem: D (ft) ee 5 (ft) Δ (ft) L (ft).5 ν. 5 (ft^/s) g 3. (m/s^) ΣKm.5.., i.e., ΣKm 7.5 For the pump: a.9 b 3. c 7. Using the Swamee-Jain equation.5 fsj, r for the friction factor: 5.7 log r Q ee Q L the sstem equation becomes: hp Δ ΣKm fsj, [] D π ν D D π g D The pump equation is: hp a b Q c Q [] A graphical analsis shows the solution as the intersection of the sstem and the pump curves, i.e., hp Q Δ Q π g D ΣKm ee fsj, D Q π ν D L D hp Q a b Q c Q 3 / 3

14 Oct 7::3 - GraphsBasics5_ForPrinting.sm hp hp Notice that, even though we defined hp and hp as functions of Q, when using the D plot, we need to define them as functions of. The eact solution can be found b solving the equation: hp(q)=hp(q) solve hp Q hp Q, Q,, 3. EXAMPLE - More eamples of three-dimensional graphs - surfaces: Use the option "3D" in the "Functions" palette, and enter the function f(,) in the placeholder. The result is a 3D surface, in this case, a plane. The original plot is shown above. Use the "Rotate" option in the "Plot" palette to change the surface view. / 3

15 Oct 7::3 - GraphsBasics5_ForPrinting.sm This figure uses the option "Graph b Lines" in the "Plot" palette (default). This figure uses the option "Graph b Points" in the "Plot" palette Use the "Move" option in the "Plot" palette to move the location of the origin in the graphics window. Use the "Scale" option in the "Plot" palette, clic on the graph, and drag the mouse over it to oom in or out. Note: Use the "Refresh" option in the "Plot" menu to recover the original version of an plot. EXAMPLE 5 - The following eamples use more comple 3D surfaces: 5 / 3

16 sin 5 The tpe of 3D graphs of surfaces produced b SMath Studio are referred to as wireframe plots. EXAMPLE - The following eamples show more than one surface plot together in 3D: Oct 7::3 - GraphsBasics5_ForPrinting.sm / 3

17 Oct 7::3 - GraphsBasics5_ForPrinting.sm sin sin EXAMPLE 7 - Drawing a surface and a space curve together: Use the "Multiple values" option in the "Plot" palette, and enter the equation of the surface (e.g., +) and the matri that represents the space curve (e.g., M) M EXAMPLE - Plotting space curves: t n n,.9.. length t In this eample two straight lines in 3D are produced b using linear parametric equations. Vector t is the parameter, and the coordinates of the two curves are given b (,,) and (,,). These lines are represented b the matrices P and Q, respectivel. for.. n 5 t t t for.. n 3 t t t P augment,, Q augment,, 7 / 3

18 Oct 7::3 - GraphsBasics5_ForPrinting.sm P Individual plots of curves given b matrices P and Q. Q <--- Join plot of lines given b matrices P and Q P Q Plot specifications using SMath Studio.9 - Release SMath Studio.9 - Release includes the abilit of specifiing four different tpes of smbols for the graphs, modifing the sie of the smbols, and select the color of the smbol. This is accomplished b building a matri that includes the coordinates (,), the character, its sie, and the color, in that order. * Characters: ou can select for plotting the characters: "" "*" "." "o" * Sie: given in piels, e.g., 5,,,, etc. * Colors available (figure below split for printing purposes): / 3

19 Oct 7::3 - GraphsBasics5_ForPrinting.sm You can write them in lower case lettes, as shown above, or all in upper case letters, or combinations of upper and lower case letters. For eample, ou could write in our color specification "darblue", "DARKBLUE", or "DarBlue", and the result would be the same. Additionall, ou can have spaces in the color specification which will be ignored b SMath Studio in producing the output to the graphic canvas. For eample, ou can write "Dar Blue", and it will be interpreted as "darblue". NOTE: An color combination, such as "LightRed", not defined above, will produce, b default, the color blac. EXAMPLE 9 - Plotting a single character or a character string: You can define a row vector with the properties: (,,char,sie,color) to write a single character or a string of characters, e.g., ch 5 "α" 5 "Blue" ch 5 "CEE 35 Hdraulics"5 "red" 9 / 3

20 Oct 7::3 - GraphsBasics5_ForPrinting.sm α CEE 35 Hdraulics ch - - ch EXAMPLE - Plotting a matri of characters: It is possible to produce a plot including a variet of single characters, or character strings, b creating a matri in which each row represents an individual string output, e.g., ch3 5 5 "Gilberto E. Urro" "CEE 35 - Hdraulics" "Fall Semester " "Using SMath Studio" "Utah State Universit" "Red" "Dar Blue" "Green" "Dar Green" "Blue" Gilberto E. Urro CEE 35 - Hdraulics Fall Semester ch3 Using SMath Studio Utah State Universit A program to plot single data sets with different smbols, sies, and colors The following program was made available b Prof. Radovan Omorjan in the SMath Studio Forum. The program taes vectors of values (,) and creates a plot matri for the data using a specified character (char), sie, and color. This program operates onl in SMath Studio version.9 release, and later. The epression for the program, or function, is: plotg,, char, sie, color n length plot augment,, char, sie, color for i.. n plot stac plot, augment,, char, sie, color i i plot / 3

21 Oct 7::3 - GraphsBasics5_ForPrinting.sm EXAMPLE - Plotting data (,) with function "plotg" The following graphs show the plotting of the function = sin(), with values of between and, in increments of.5, using different smbols, sies, and colors. The data is generated here:,.5.. n length n for.. n sin The plots are shown below: plot plotg,, ".", 5, "Light Green" plot plotg,, "+", 35, "Magenta" plot plot plot3 plotg,, "", 5, "Dar Blue" plot plotg,, "o",, "Violet" plot3 -.5 plot EXAMPLE - Plotting various data (,) with function "plotg" The following graphs show the plotting of the function = sin(), with values of between and, in increments of.5, using different smbols, sies, and colors. The data is generated here:,.5.. n length n / 3

22 for r Oct 7::3 - GraphsBasics5_ForPrinting.sm.. n sin sin cos 3 The individual plots of (,), (,), and (,r) are given b sy, sz, and sr: sy sz sr plotg,, ".", 5, "Blue" plotg,, "o",, "Red" plotg, r, "", 5, "Magenta" The following matrices combine plots of various data sets: MYZ stac sy, sz MZR stac sz, sr MYR stac sy, sr MYZR stac sy, sz, sr Some plots based on the matrices, and the original functions used to produce the matri data, are shown net: MYZ MZR MYR -.5 MYZR / 3

23 Oct 7::3 - GraphsBasics5_ForPrinting.sm MYZ sin - - MYZ sin sin 3 / 3